Ferromagnetic (Ga,Mn)As Layers and ... - OPUS Würzburg

58 4. Finite Element Simulations of Strain Relaxation
1 .0 x 1 0 -3
8 .0 x 1 0 -4
6 .0 x 1 0 -4
4 .0 x 1 0 -4
S tra in
2 .0 x 1 0 -4
0 .0
-2 .0 x 1 0 -4
-4 .0 x 1 0 -4
e (1 1 0 )
e z
e x y
e x = e y
e x (1 0 0 s trip e )
e z (1 0 0 s trip e )
-6 .0 x 1 0 -4
-8 .0 x 1 0 -4
-1 .0 x 1 0 -3
2 4 6 8 1 0
R e la tiv e W id th
Figure 4.8: Simulation of the average strain in a (Ga,Mn)As stripe aligned along the [ 110 ]
crystal direction. The solid lines represent the strain in the cubic directions (blue: e x = e y ,
red: e z ), the shear strain (green: e xy ), and the calculated relaxation perpendicular to the
stripe axis (black: e [110] ). The dotted lines are the simulation results for an identical stripe
aligned along [100], taken from Fig. 4.4.
By comparing Eqn. (4.19) and Eqn. (4.20), we can now identify:
e [110]
= 1 2 (e x − 2e xy + e y ) (4.21)
e [110] = 1 2 (e x + 2e xy + e y ) (4.22)
For the given material parameters of the stipe, the strain in [1¯10] is equal to
−1.5 · 10 −3 , as no strain relaxation takes place in this direction. The strain e [110]
can be directly compared to e x for a stripe along [100] to determine the difference in
perpendicular relaxation for both geometries. We note that in the rotated matrix E ′ ,
the shear strain vanishes, as we would expect for a stripe aligned with the coordinate
axes.
The strain simulation shown in Fig. 4.8 was compiled by varying the relative width
of a (Ga,Mn)As stripe with identical parameters as discussed in Section 4.3.1.
By comparing the relaxation perpendicular to the stripe, e [110] , with the the corresponding
e x (dotted blue line) of the nonrotated stripe, we can immediately see,
that we achieve a lesser degree of relaxation in this geometry for otherwise identical
structures. This fact is also evident in the larger value of e z in the rotated stripe. The